Cross tabulation

Assuming we have a categorical independent variable (IV) and a categorical dependent variable (DV):

Cross tabulation: Step 1

Start by calculating the number of observations with each value of each category:

Cross tabulation: Step 2

Then, calculate the proportion/percentage of observations among each value of the IV.

If the independent variable is in the columns, then the columns should sum to 100%.

If the independent variable is in the rows, then the rows should sum to 100%.

Cross tabulation: interpretation

Look at what happens to the DV at different values of the IV. If your variables are ordinal, you should be able to identify a direction of the effect.

The proportion of “Yes” values decreases as the IV goes from lower to higher, so this is a negative or inverse relationship.

Cross tabulation: notes

• Key rule: always calculate percentages or proportions by categories of the independent variable.

  • This allows you to compare groups that are different sizes.

• If one or both variables are interval-level, you can bin them in order to use them in a cross tab. For instance, you could separate an interval like into a series of age ranges.

Cross tabulation: example

Hypothesis: in a comparison of individuals, independents are less likely to turn out to vote compared to people who support one party or another.

Cross tabulation: example

Are these results generally consistent with my hypothesis?

Row/Column percentages

What happens if I calculate % among the values of the DV?

Here’s the relationship between education and voter turnout with % calculated on education level:

Row/Column percentages

What happens if I calculate % among the values of the DV?

Here’s the relationship between education and voter turnout with % calculated across voter turnout

Row/Column percentages

Either of these tables might be a valid way to look at these data, but they answer slightly different questions:

Mean Comparison

When we have interval level outcome and a categorical independent variable, we can group each observation by values of the IV and then calculate the mean across each group.

For instance I want to examine the relationship between national wealth and carbon emissions. My hypothesis is that wealthier nations will have more emissions compared to poorer nations.

Mean Comparison

GDP data has been grouped into five categories, so now I just need to calculate the average of CO2 emissions within each group of the ordinal IV:

Mean Comparison

Here again, the relationship can be easier to conceptualize if we plot it.

Curvilinear Relationships

A relationship like this will rarely be perfectly straight, so “linearity” and “curvilinearity” are partly a matter of degree, but there are some cases where there is a clear “U” shape to the relationship:

Research Design

Rival Explanations

• How can we distinguish correlation from causation?

• This process inevitably requires us to consider rival explanations for an observed relationship:

  • For instance: if I find that social media use correlated with a lower likelihood of turning out to vote, I might ask whether age is a confounder that could explain this correlation.

Confounding

What I want to show is that Fox News viewership is cases a decreased chance of getting a Covid vaccine.

Confounding

There’s a correlation, but I’m concerned this relationship is spurious because I know that things like existing political views are already correlated with media consumption, and those might explain any correlation I see here:

Confounding

What if I could randomly assign people to watch Fox News? Random assignment would ensure that nothing is correlated with Fox news viewership.

Experiments

Experiments

Experiments: Field Experiments

Field experiments can lessen the external validity problem by using random assignment in the field.

Experiments: Field Experiments

From: GERBER, A. S., GREEN, D. P., & LARIMER, C. W. (2008). Social Pressure and Voter Turnout: Evidence from a Large-Scale Field Experiment. American Political Science Review, 102(1), 33–48. doi:10.1017/S000305540808009X

Experiments: Natural Experiments

Field experiments can face fewer external validity problems, but some things still can’t be experimentally manipulated.

Natural experiments use “quasi” randomization or “randomization by nature” where treatments are assigned more-or-less randomly.

Experiments: Natural Experiments

Experiments: Natural Experiments

Other sources of quasi randomization include:

Observational Research

Observational research doesn’t allow us to easily account for rival explanations. So how can we cope?